S3.3StatisticsCore

Cumulative Frequency

A cumulative frequency graph plots running totals against the upper class boundaries. The resulting S-shaped curve lets you read off the median, quartiles and the number of values below any given amount.

25 min Video by Zeeshan Zamurred Representations of Data

Edexcel Statistics Y1 — Representations of Data playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Build a cumulative frequency table
Plot a cumulative frequency graph
Read off the median and quartiles
Estimate how many values lie below a value

Cumulative frequency

The $cumulative frequency$ at a point is the running total of all frequencies up to and including that class. Plot it against the $upper class boundary$ of each class and join the points with a smooth curve.

Tip — Always plot cumulative frequency against the UPPER boundary of each class — a common slip is to use the midpoint.

Reading off statistics

From the curve, read across from the relevant cumulative frequency to find a value. For $n$ data points, the median is at $\frac{n}{2}$ , $Q_{1}$ at $\frac{n}{4}$ and $Q_{3}$ at $\frac{3 n}{4}$ on the cumulative axis.

median at \frac{n}{2}, Q_{1} at \frac{n}{4}, Q_{3} at \frac{3 n}{4}

Read across from these heights, then down to the value axis.

Values below a number

To estimate how many data points are below a given value, read up from that value to the curve and across to the cumulative frequency. Subtract from $n$ for “how many above”.

1Below 50: 30. Above:

80 - 30

Answer50 values

Formula recap

cf = running total of frequencies

Cumulative frequency.

plot cf vs upper class boundary

How to plot.

median \frac{n}{2}, Q_{1} \frac{n}{4}, Q_{3} \frac{3 n}{4}

Reading quartiles.

Common mistakes to avoid

Plotting cumulative frequency against the class midpoint.

Use the UPPER class boundary.

Reading the median at (n+1)/2 on a cumulative frequency graph.

For a large grouped data curve, use n/2.

Key takeaways

Cumulative frequency is a running total; plot it vs upper class boundaries.
Read median, Q₁, Q₃ at n/2, n/4, 3n/4 on the cumulative axis.
Read across/down to estimate values below a given number.

Test yourself

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